(October 9, 2011) Modular curves, raindrops through kaleidoscopes Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ • H as homogeneous space for SL2(R) • The simplest non-abelian quotient SL2(Z)nH • Non-abelian solenoid: raindrop through a kaleidoscope • Enabling the action of SL2(R) Solenoids are limits of one-dimensional things, circles. As complicated as the solenoids might be, all the limitands are the same, just circles. Hoping for a richer supply of phenomena, we should wonder about limits of two-dimensional things (surfaces). It was convenient that all the circles in the solenoids were uniformized compatibly, that is, were compatibly expressed as quotients of R. For ease of description, we choose to look at surfaces X which are quotients of [1] the upper half-plane H in C, with quotient mappings H ! X that fit together compatibly. The real line R, being a group, is a homogeneous space in the sense that (of course) it acts transitively on itself. The upper half-plane H is not reasonably a group itself, but is acted upon by SL2(R) (2-by-2 real matrices with determinant 1) acting by so-called linear fractional transformations a b az + b (z) = c d cz + d We verify that this action is transitive, making H a homogeneous space for the group SL2(R). The oddity of the action can be put in a larger context, which we will do a bit later. Modular curves are quotients H ! ΓnH where Γ varies among certain finite-index subgroups of SL2(Z), the group of 2-by-2 integer matrices with determinant 1. They are called modular [2] for historical reasons discussed earlier. They are curves in the sense that they are complex one-dimensional (though real two- dimensional). 2 We will see that SL2(Z)nH, the simplest explicit case, is topologically S with a point missing. However, when we keep track of geometry consistent with the relevant group action, the missing point is infinitely far away, so the shape is not a round sphere, but stretched out like a raindrop. The hyperbolic geometry appropriate to the upper half-plane will be discussed briefly in the next chapter. Some of the discussion here will seem peculiarly specific, or peculiarly idiosyncratic, especially by comparison to the ease with which we manipulate R and Z. The contrast is partly explained by the fact that two- dimensional objects with non-abelian group actions are genuinely more complicated. Indeed, at some future point, we may decide (with hindsight) that as groups or geometric objects R and R=Z are misleading. And why not look at quotients of C (or R2) to make surfaces? Indeed, the quotients of C by lattices are elliptic curves, certainly worthy of study. We choose a different course with its own interest. [1] The Uniformization Theorem in complex analysis asserts that a compact, connected, Riemann surface (that is, a 1 compact, connected, one-dimensional complex manifold) is either complex projective 1-space P (a.k.a. the Riemann sphere), or is C=Λ for a lattice Λ, or is a quotient ΓnD of the unit disk D by a suitable group Γ ⊂ GL2(C). (It is easy to check that the map z ! (z + i)=(iz + 1) is an isomorphism of the disk to the upper half-plane, so uniformizing by one is equivalent to uniformizing by the other.) In this last (and very interesting) case of uniformization by the disk or half-plane, however, the groups Γ rarely can be described sufficiently tangibly for our present purposes. [2] For the same historical reasons, SL2(Z) is sometimes called the modular group. By now this is an anachronism. 1 Paul Garrett: Modular curves, raindrops through kaleidoscopes (October 9, 2011) 1. H as homogeneous space for SL2(R) Before trying to uniformize surfaces, we must explain the structure of the half-plane H as homogeneous space, that is, as a space acted upon transitively by a group. When uniformizing circles as R ! R=Z, the group R acting transitively on such a circle is not very different from the circle itself. Indeed, circles R=Z are groups themselves, since Z is normal in R, which is unavoidable since R is abelian. By contrast, the upper half-plane H = fz = x + iy : y > 0g ⊂ C does not have any reasonable group structure itself. Luckily, the group a b G = SL ( ) = f : real matrices with ad − bc = 1g 2 R c d acts on H with the linear fractional transformation action [3] a b az + b (z) = c d cz + d and we have: [4] [1.0.1] Claim: The group SL2(R) stabilizes H and acts transitively on it. In particular, p 1 x y 0 p1 (i) = x + iy (for x 2 R, y > 0) 0 1 0 y a b Further, for g = 2 SL ( ) and z 2 H c d 2 R Im z Im g(z) = jcz + dj2 Proof: The first formula is clear. The second formula would imply that the upper half-plane is stabilized. Compute directly: a b az + b az + b (az + b)(cz + d) − (az + b)(cz + d) 2i · Im( (z)) = − = c d cz + d cz + d jcz + dj2 adz − bcz − bcz + adz z − z = = jcz + dj2 jcz + dj2 since ad − bc = 1. === [3] This action can easily (if awkwardly) be discussed in a completely ad hoc fashion, but, in fact, arises as an artifact 1 (in coordinates) of a natural action of GL2(C) on complex projective space P . This situation itself is a very special n case of actions of GL(n + 1; C) (complex invertible matrices of size n + 1) on projective n-space P . This broader context, as well as action of subgroups stabilizing complex n-balls, will be discussed just a little later. [4] In fact, every holomorphic automorphism of H is given by an element of SL2(R). This follows from Schwarz' lemma (on the disk), which allows us to deduce that an automorphism of the disk fixing 0 is a rotation. 2 Paul Garrett: Modular curves, raindrops through kaleidoscopes (October 9, 2011) [1.0.2] Remark: The extra information about how the imaginary part transforms will be useful in determining a fundamental domain just below. Since SL2(R) acts transitively on H, we can express H as a quotient of SL2(R). For example, [5] [1.0.3] Claim: The isotropy group in SL2(R) of the point i 2 H is the special orthogonal group cos θ sin θ SO(2) = fg 2 SL ( ): g> · g = 1 g = f : θ 2 g 2 R 2 − sin θ cos θ R Proof: For real a; b; c; d, the equation (ai + b)=(ci + d) = i gives ai + b = −c + id, so a = d and c = −b. The determinant condition ad − bc = 1 gives a2 + b2 = 1, which we can reparametrized via trigonometric functions as indicated. === [1.0.4] Corollary: We have an isomorphism of SL2(R)-spaces SL2(R)=SO(2) ≈ H via g · SO(2) ! g(i) [1.0.5] Remark: The last claim and its corollary had only a boring analogue in the case of the transitive action of R on itself, since all isotropy groups were trivial. [1.0.6] Remark: Yes, quotients such as SL2(R)=SO(2) have more structure than just topological, and these will be relevant. [1.0.7] Remark: It will eventually become clear that the effect of taking the quotient of SL2(R) by SO(2) is a hindrance, and that we should prefer to consider the three-dimensional group SL2(R) and its quotients ΓnSL2(R), rather than SL2(R)=SO(2) and its quotients ΓnSL2(R)=SO(2) ≈ ΓnH for several reasons. However, in the short term, and to connect with (and exploit) historical artifacts, we do treat H and its quotients ΓnH. 2. The simplest non-abelian quotient SL2(Z)nH [6] We will make surfaces as quotients ΓnH of the half-plane H by subgroups Γ of G = SL2(R) such that the quotient ΓnH is reasonably small. [7] The simplest beginning choice is Γ = SL2(Z) = f2-by-2 integer matrices with determinant 1g Both for use just below and to show that SL2(Z) is a large group, we note: [5] The choice of corner in which to put the − sin θ does not matter much in the larger scheme of things, and often the opposite choice is made, but there are some reasons one might make the present choice. Still, it doesn't really matter. [6] As in the case of the solenoids, the local topology of quotients is simplest for quotients by discrete subgroups. See the appendix where configurations ΓnG=K are considered, for Γ discrete in a topological group G, and K a compact subgroup of G. [7] The Uniformization Theorem makes compact (connected) surfaces, but our explicitly-constructed surfaces will not be compact. This creates non-trivial issues in many regards, but the relative simplicity of the description of the groups Γ makes these other complications acceptable. 3 Paul Garrett: Modular curves, raindrops through kaleidoscopes (October 9, 2011) a b [2.0.1] Claim: Given relatively prime integers c; d, there are integers a; b such that 2 Γ. c d Proof: From basic number theory we know that there are integers m; n such that greatest common divisor c; d = m · c + n · d Here the greatest common divisor is 1, and take a = n, b = −m, so ad − bc = 1. === To be able to draw a picture of the quotient, we take an archaic [8] approach which nevertheless succeeds in this case.